Abstract
Patterson’s work [1] on the invariants of the rational Bézier paths may be extended to permit weight vectors of mixed-sign [2]. In this more general situation, in addition to Patterson’s continuous invariants, a discrete sign-pattern invariant is required to distinguish path geometry. The author’s derivation of the invariants differs from that of Patterson’s and extends naturally to the rational Bézier surfaces. In this paper it is shown that 13 continuous invariant functions and a discrete, sign-pattern, invariant exist for the bi-cubic surfaces. Explicit functional forms of the invariant functions for the bi-cubics are obtained. The results are viewed from the perspective of the Fundamental Theorem on invariants for Lie groups.
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References
Patterson, R.R.: Projective transformations of the parameter of a Bernstein-Bézier curve. ACM Trans. on Graphics 4(4), 276–290 (1985)
Bez, H.E.: Generalised invariant-geometry conditions for the rational Bézier paths. International Journal of Computer Mathematics (to appear)
Farin, G.: NURBS from Projective Geometry to Practical Use, 2nd edn. A.K.Peters (1999)
Bez, H.E.: Invariant-geometry conditions for the rational bi-quadratic Bézier surfaces (submitted for publication) (2008)
Bez, H.E.: Bounded domain bi-quadratic rational parametrisations of Dupin cyclides. International Journal of Computer Mathematics 85(7), 1097–1111 (2008)
Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999)
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© 2009 Springer-Verlag Berlin Heidelberg
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Bez, H.E. (2009). The Invariant Functions of the Rational Bi-cubic Bézier Surfaces. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds) Mathematics of Surfaces XIII. Mathematics of Surfaces 2009. Lecture Notes in Computer Science, vol 5654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03596-8_4
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DOI: https://doi.org/10.1007/978-3-642-03596-8_4
Publisher Name: Springer, Berlin, Heidelberg
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